Mid-point Formula

The midpoint formula is a fundamental concept in coordinate geometry that allows us to determine the exact middle point of a line segment connecting two given points in a Cartesian coordinate system. It essentially provides the coordinates of the point that lies exactly halfway between the two endpoints of the segment.

Formulae

The midpoint formula is given by:

Midpoint = $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Where:

• $(x_1, y_1)$ are the coordinates of the first endpoint.
• $(x_2, y_2)$ are the coordinates of the second endpoint.

Examples

Let’s illustrate the midpoint formula with a few examples.

Example-1: Find the midpoint of the line segment with endpoints A(2, 3) and B(4, 7).

Solution:

Using the midpoint formula:

Midpoint = $\left(\frac{2 + 4}{2}, \frac{3 + 7}{2}\right)$

Midpoint = $\left(\frac{6}{2}, \frac{10}{2}\right)$

Midpoint = (3, 5)

Therefore, the midpoint of the line segment AB is (3, 5).

Example-2: Determine the midpoint of the line segment with endpoints C(-1, -2) and D(5, 0).

Solution:

Using the midpoint formula:

Midpoint = $\left(\frac{-1 + 5}{2}, \frac{-2 + 0}{2}\right)$

Midpoint = $\left(\frac{4}{2}, \frac{-2}{2}\right)$

Midpoint = (2, -1)

Therefore, the midpoint of the line segment CD is (2, -1).

Common mistakes by students

Here are some common mistakes students make when using the midpoint formula:

• Incorrect Order of Operations: Forgetting to add the x-coordinates and y-coordinates *before* dividing by 2.
• Mixing up Coordinates: Swapping the x and y coordinates in the formula.
• Arithmetic Errors: Making simple calculation mistakes, especially with negative numbers.
• Confusing with Distance Formula: Mixing up the midpoint formula with the distance formula.

Real Life Application

The midpoint formula has practical applications in various real-world scenarios:

• Mapping and Navigation: Determining the center of a geographical area or the average position of two locations.
• Computer Graphics: Finding the center of a shape or object for rendering and animation.
• Engineering: Calculating the centroid (center of mass) of a structure or component.

Fun Fact

The midpoint formula is a direct consequence of the concept of averages. The x-coordinate and y-coordinate of the midpoint are simply the average of the x-coordinates and y-coordinates of the endpoints, respectively. This links the formula to concepts you’ve likely already encountered in statistics and data analysis!

Recommended YouTube Videos for Deeper Understanding

Q.1 Given points A(2, 4) and B(6, 8), find the midpoint M of the line segment AB.

Check Solution

Ans: A

Using the midpoint formula, M = ((2+6)/2, (4+8)/2) = (4, 6).

Q.2 The midpoint of a line segment is (3, -1). One endpoint is (5, 2). What are the coordinates of the other endpoint?

Check Solution

Ans: A

Let the other endpoint be (x, y). Then, ((x+5)/2, (y+2)/2) = (3, -1). Solving for x and y, we get x = 1 and y = -4.

Q.3 If the endpoints of a diameter of a circle are (1, 5) and (7, 1), what are the coordinates of the center of the circle?

Check Solution

Ans: B

The center of the circle is the midpoint of the diameter. Using the midpoint formula, ((1+7)/2, (5+1)/2) = (4, 3).

Q.4 The endpoints of a line segment AB are A(-3, 7) and B(5, -1). If point C divides the segment AB in half, what are the coordinates of C?

Check Solution

Ans: C

Point C is the midpoint of AB. Applying the midpoint formula, C = ((-3+5)/2, (7-1)/2) = (1, 3).

Q.5 A line segment has endpoints P(a, b) and Q(c, d). The midpoint of the line segment is located at (4, 7). If a = 2 and d = 10, what is the value of b + c?

Check Solution

Ans: C

Using the midpoint formula: ((a+c)/2, (b+d)/2) = (4, 7). Substituting a=2 and d=10, we get ((2+c)/2, (b+10)/2) = (4, 7). Solving for the x-coordinate: (2+c)/2 = 4 => c = 6. Solving for the y-coordinate: (b+10)/2 = 7 => b = 4. Therefore, b+c = 4+6 = 10.

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